Optimal. Leaf size=172 \[ -\frac {3 b^2 e^3 (c+d x)^2}{32 d}+\frac {b^2 e^3 (c+d x)^4}{32 d}+\frac {3 b e^3 (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{16 d}-\frac {b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{8 d}-\frac {3 e^3 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{4 d} \]
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Rubi [A]
time = 0.17, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {5859, 12, 5776,
5812, 5783, 30} \begin {gather*} \frac {e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{4 d}-\frac {b e^3 \sqrt {(c+d x)^2+1} (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )}{8 d}+\frac {3 b e^3 \sqrt {(c+d x)^2+1} (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )}{16 d}-\frac {3 e^3 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{32 d}+\frac {b^2 e^3 (c+d x)^4}{32 d}-\frac {3 b^2 e^3 (c+d x)^2}{32 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 5776
Rule 5783
Rule 5812
Rule 5859
Rubi steps
\begin {align*} \int (c e+d e x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^2 \, dx &=\frac {\text {Subst}\left (\int e^3 x^3 \left (a+b \sinh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^3 \text {Subst}\left (\int x^3 \left (a+b \sinh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{4 d}-\frac {\left (b e^3\right ) \text {Subst}\left (\int \frac {x^4 \left (a+b \sinh ^{-1}(x)\right )}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{2 d}\\ &=-\frac {b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{8 d}+\frac {e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{4 d}+\frac {\left (3 b e^3\right ) \text {Subst}\left (\int \frac {x^2 \left (a+b \sinh ^{-1}(x)\right )}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{8 d}+\frac {\left (b^2 e^3\right ) \text {Subst}\left (\int x^3 \, dx,x,c+d x\right )}{8 d}\\ &=\frac {b^2 e^3 (c+d x)^4}{32 d}+\frac {3 b e^3 (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{16 d}-\frac {b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{8 d}+\frac {e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{4 d}-\frac {\left (3 b e^3\right ) \text {Subst}\left (\int \frac {a+b \sinh ^{-1}(x)}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{16 d}-\frac {\left (3 b^2 e^3\right ) \text {Subst}(\int x \, dx,x,c+d x)}{16 d}\\ &=-\frac {3 b^2 e^3 (c+d x)^2}{32 d}+\frac {b^2 e^3 (c+d x)^4}{32 d}+\frac {3 b e^3 (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{16 d}-\frac {b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{8 d}-\frac {3 e^3 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{4 d}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 170, normalized size = 0.99 \begin {gather*} \frac {e^3 \left (-3 b^2 (c+d x)^2+\left (8 a^2+b^2\right ) (c+d x)^4+2 a b (c+d x) \left (3-2 (c+d x)^2\right ) \sqrt {1+(c+d x)^2}-6 a b \sinh ^{-1}(c+d x)+2 b (c+d x) \left (8 a (c+d x)^3+3 b \sqrt {1+(c+d x)^2}-2 b (c+d x)^2 \sqrt {1+(c+d x)^2}\right ) \sinh ^{-1}(c+d x)+b^2 \left (-3+8 (c+d x)^4\right ) \sinh ^{-1}(c+d x)^2\right )}{32 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 3.54, size = 222, normalized size = 1.29
method | result | size |
default | \(\frac {e^{3} \left (d x +c \right )^{4} a^{2}}{4 d}-\frac {e^{3} b^{2} \left (-16 \left (\cosh ^{2}\left (2 \arcsinh \left (d x +c \right )\right )\right ) \arcsinh \left (d x +c \right )^{2}+8 \sinh \left (2 \arcsinh \left (d x +c \right )\right ) \cosh \left (2 \arcsinh \left (d x +c \right )\right ) \arcsinh \left (d x +c \right )+32 \cosh \left (2 \arcsinh \left (d x +c \right )\right ) \arcsinh \left (d x +c \right )^{2}-32 \arcsinh \left (d x +c \right ) \sinh \left (2 \arcsinh \left (d x +c \right )\right )-2 \left (\cosh ^{2}\left (2 \arcsinh \left (d x +c \right )\right )\right )+8 \arcsinh \left (d x +c \right )^{2}+16 \cosh \left (2 \arcsinh \left (d x +c \right )\right )+1\right )}{256 d}+\frac {2 e^{3} a b \left (\frac {\left (d x +c \right )^{4} \arcsinh \left (d x +c \right )}{4}-\frac {\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{16}+\frac {3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{32}-\frac {3 \arcsinh \left (d x +c \right )}{32}\right )}{d}\) | \(222\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1477 vs.
\(2 (150) = 300\).
time = 0.38, size = 1477, normalized size = 8.59 \begin {gather*} \frac {{\left ({\left (8 \, a^{2} + b^{2}\right )} d^{4} x^{4} + 4 \, {\left (8 \, a^{2} + b^{2}\right )} c d^{3} x^{3} + 3 \, {\left (2 \, {\left (8 \, a^{2} + b^{2}\right )} c^{2} - b^{2}\right )} d^{2} x^{2} + 2 \, {\left (2 \, {\left (8 \, a^{2} + b^{2}\right )} c^{3} - 3 \, b^{2} c\right )} d x\right )} \cosh \left (1\right )^{3} + 3 \, {\left ({\left (8 \, a^{2} + b^{2}\right )} d^{4} x^{4} + 4 \, {\left (8 \, a^{2} + b^{2}\right )} c d^{3} x^{3} + 3 \, {\left (2 \, {\left (8 \, a^{2} + b^{2}\right )} c^{2} - b^{2}\right )} d^{2} x^{2} + 2 \, {\left (2 \, {\left (8 \, a^{2} + b^{2}\right )} c^{3} - 3 \, b^{2} c\right )} d x\right )} \cosh \left (1\right )^{2} \sinh \left (1\right ) + 3 \, {\left ({\left (8 \, a^{2} + b^{2}\right )} d^{4} x^{4} + 4 \, {\left (8 \, a^{2} + b^{2}\right )} c d^{3} x^{3} + 3 \, {\left (2 \, {\left (8 \, a^{2} + b^{2}\right )} c^{2} - b^{2}\right )} d^{2} x^{2} + 2 \, {\left (2 \, {\left (8 \, a^{2} + b^{2}\right )} c^{3} - 3 \, b^{2} c\right )} d x\right )} \cosh \left (1\right ) \sinh \left (1\right )^{2} + {\left ({\left (8 \, a^{2} + b^{2}\right )} d^{4} x^{4} + 4 \, {\left (8 \, a^{2} + b^{2}\right )} c d^{3} x^{3} + 3 \, {\left (2 \, {\left (8 \, a^{2} + b^{2}\right )} c^{2} - b^{2}\right )} d^{2} x^{2} + 2 \, {\left (2 \, {\left (8 \, a^{2} + b^{2}\right )} c^{3} - 3 \, b^{2} c\right )} d x\right )} \sinh \left (1\right )^{3} + {\left ({\left (8 \, b^{2} d^{4} x^{4} + 32 \, b^{2} c d^{3} x^{3} + 48 \, b^{2} c^{2} d^{2} x^{2} + 32 \, b^{2} c^{3} d x + 8 \, b^{2} c^{4} - 3 \, b^{2}\right )} \cosh \left (1\right )^{3} + 3 \, {\left (8 \, b^{2} d^{4} x^{4} + 32 \, b^{2} c d^{3} x^{3} + 48 \, b^{2} c^{2} d^{2} x^{2} + 32 \, b^{2} c^{3} d x + 8 \, b^{2} c^{4} - 3 \, b^{2}\right )} \cosh \left (1\right )^{2} \sinh \left (1\right ) + 3 \, {\left (8 \, b^{2} d^{4} x^{4} + 32 \, b^{2} c d^{3} x^{3} + 48 \, b^{2} c^{2} d^{2} x^{2} + 32 \, b^{2} c^{3} d x + 8 \, b^{2} c^{4} - 3 \, b^{2}\right )} \cosh \left (1\right ) \sinh \left (1\right )^{2} + {\left (8 \, b^{2} d^{4} x^{4} + 32 \, b^{2} c d^{3} x^{3} + 48 \, b^{2} c^{2} d^{2} x^{2} + 32 \, b^{2} c^{3} d x + 8 \, b^{2} c^{4} - 3 \, b^{2}\right )} \sinh \left (1\right )^{3}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2} + 2 \, {\left ({\left (8 \, a b d^{4} x^{4} + 32 \, a b c d^{3} x^{3} + 48 \, a b c^{2} d^{2} x^{2} + 32 \, a b c^{3} d x + 8 \, a b c^{4} - 3 \, a b\right )} \cosh \left (1\right )^{3} + 3 \, {\left (8 \, a b d^{4} x^{4} + 32 \, a b c d^{3} x^{3} + 48 \, a b c^{2} d^{2} x^{2} + 32 \, a b c^{3} d x + 8 \, a b c^{4} - 3 \, a b\right )} \cosh \left (1\right )^{2} \sinh \left (1\right ) + 3 \, {\left (8 \, a b d^{4} x^{4} + 32 \, a b c d^{3} x^{3} + 48 \, a b c^{2} d^{2} x^{2} + 32 \, a b c^{3} d x + 8 \, a b c^{4} - 3 \, a b\right )} \cosh \left (1\right ) \sinh \left (1\right )^{2} + {\left (8 \, a b d^{4} x^{4} + 32 \, a b c d^{3} x^{3} + 48 \, a b c^{2} d^{2} x^{2} + 32 \, a b c^{3} d x + 8 \, a b c^{4} - 3 \, a b\right )} \sinh \left (1\right )^{3} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} {\left ({\left (2 \, b^{2} d^{3} x^{3} + 6 \, b^{2} c d^{2} x^{2} + 2 \, b^{2} c^{3} - 3 \, b^{2} c + 3 \, {\left (2 \, b^{2} c^{2} - b^{2}\right )} d x\right )} \cosh \left (1\right )^{3} + 3 \, {\left (2 \, b^{2} d^{3} x^{3} + 6 \, b^{2} c d^{2} x^{2} + 2 \, b^{2} c^{3} - 3 \, b^{2} c + 3 \, {\left (2 \, b^{2} c^{2} - b^{2}\right )} d x\right )} \cosh \left (1\right )^{2} \sinh \left (1\right ) + 3 \, {\left (2 \, b^{2} d^{3} x^{3} + 6 \, b^{2} c d^{2} x^{2} + 2 \, b^{2} c^{3} - 3 \, b^{2} c + 3 \, {\left (2 \, b^{2} c^{2} - b^{2}\right )} d x\right )} \cosh \left (1\right ) \sinh \left (1\right )^{2} + {\left (2 \, b^{2} d^{3} x^{3} + 6 \, b^{2} c d^{2} x^{2} + 2 \, b^{2} c^{3} - 3 \, b^{2} c + 3 \, {\left (2 \, b^{2} c^{2} - b^{2}\right )} d x\right )} \sinh \left (1\right )^{3}\right )}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} {\left ({\left (2 \, a b d^{3} x^{3} + 6 \, a b c d^{2} x^{2} + 2 \, a b c^{3} - 3 \, a b c + 3 \, {\left (2 \, a b c^{2} - a b\right )} d x\right )} \cosh \left (1\right )^{3} + 3 \, {\left (2 \, a b d^{3} x^{3} + 6 \, a b c d^{2} x^{2} + 2 \, a b c^{3} - 3 \, a b c + 3 \, {\left (2 \, a b c^{2} - a b\right )} d x\right )} \cosh \left (1\right )^{2} \sinh \left (1\right ) + 3 \, {\left (2 \, a b d^{3} x^{3} + 6 \, a b c d^{2} x^{2} + 2 \, a b c^{3} - 3 \, a b c + 3 \, {\left (2 \, a b c^{2} - a b\right )} d x\right )} \cosh \left (1\right ) \sinh \left (1\right )^{2} + {\left (2 \, a b d^{3} x^{3} + 6 \, a b c d^{2} x^{2} + 2 \, a b c^{3} - 3 \, a b c + 3 \, {\left (2 \, a b c^{2} - a b\right )} d x\right )} \sinh \left (1\right )^{3}\right )}}{32 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 916 vs.
\(2 (155) = 310\).
time = 0.58, size = 916, normalized size = 5.33 \begin {gather*} \begin {cases} a^{2} c^{3} e^{3} x + \frac {3 a^{2} c^{2} d e^{3} x^{2}}{2} + a^{2} c d^{2} e^{3} x^{3} + \frac {a^{2} d^{3} e^{3} x^{4}}{4} + \frac {a b c^{4} e^{3} \operatorname {asinh}{\left (c + d x \right )}}{2 d} + 2 a b c^{3} e^{3} x \operatorname {asinh}{\left (c + d x \right )} - \frac {a b c^{3} e^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{8 d} + 3 a b c^{2} d e^{3} x^{2} \operatorname {asinh}{\left (c + d x \right )} - \frac {3 a b c^{2} e^{3} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{8} + 2 a b c d^{2} e^{3} x^{3} \operatorname {asinh}{\left (c + d x \right )} - \frac {3 a b c d e^{3} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{8} + \frac {3 a b c e^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{16 d} + \frac {a b d^{3} e^{3} x^{4} \operatorname {asinh}{\left (c + d x \right )}}{2} - \frac {a b d^{2} e^{3} x^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{8} + \frac {3 a b e^{3} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{16} - \frac {3 a b e^{3} \operatorname {asinh}{\left (c + d x \right )}}{16 d} + \frac {b^{2} c^{4} e^{3} \operatorname {asinh}^{2}{\left (c + d x \right )}}{4 d} + b^{2} c^{3} e^{3} x \operatorname {asinh}^{2}{\left (c + d x \right )} + \frac {b^{2} c^{3} e^{3} x}{8} - \frac {b^{2} c^{3} e^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{8 d} + \frac {3 b^{2} c^{2} d e^{3} x^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}{2} + \frac {3 b^{2} c^{2} d e^{3} x^{2}}{16} - \frac {3 b^{2} c^{2} e^{3} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{8} + b^{2} c d^{2} e^{3} x^{3} \operatorname {asinh}^{2}{\left (c + d x \right )} + \frac {b^{2} c d^{2} e^{3} x^{3}}{8} - \frac {3 b^{2} c d e^{3} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{8} - \frac {3 b^{2} c e^{3} x}{16} + \frac {3 b^{2} c e^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{16 d} + \frac {b^{2} d^{3} e^{3} x^{4} \operatorname {asinh}^{2}{\left (c + d x \right )}}{4} + \frac {b^{2} d^{3} e^{3} x^{4}}{32} - \frac {b^{2} d^{2} e^{3} x^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{8} - \frac {3 b^{2} d e^{3} x^{2}}{32} + \frac {3 b^{2} e^{3} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{16} - \frac {3 b^{2} e^{3} \operatorname {asinh}^{2}{\left (c + d x \right )}}{32 d} & \text {for}\: d \neq 0 \\c^{3} e^{3} x \left (a + b \operatorname {asinh}{\left (c \right )}\right )^{2} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (c\,e+d\,e\,x\right )}^3\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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